Objects, shadows and reflections not rendering in EEVEE
As Jaroslav mentioned in his comment this is an intrinsic limitation of EEVEE's rendering system, and in fact a broader shortcoming of screen space effects in general.

Since they are, as the name says, computed in screen space, only currently visible on screen objects are actually accounted for, as such, for performance reasons, as soon as an object is out of view it is discarded to save memory and rendering steps, and thus is no longer rendered.

This generally isn't a problem for direct lighting and visibility, but can have nasty side effects in inter-object relationships, like shadows, reflections, refraction, indirect lighting, among others.

There really isn't any elegant workaround or solution to this.

Luckily to minimize the effect the developers added an overscan option. 
As of commit Eevee: Implement Overscan option it can be set in the Properties Window > Render Tab > Film > Overscan.

It is an option that internally increases render size, while keeping the field of view intact, virtually creating an invisible buffer area around the camera where objects are no longer directly visible to the user but still accounted for in screen space effects that need it.


  EEVEE: Implement Overscan option
  
  This option make the internal render size larger than the output size
  in order to minimize screenspace effects disappearing at the render
  edges.
  
  The overscan size added around the render is the maximum dimension
  multiplied by the overscan percentage.


This is a workaround however and doesn't really solve the underlying limitations. While it may slightly reduce these kinds of issues, it does so at the expense of performance.

If this is still insufficient you can improve the situation by using reflection probes to get proper reflections.

If those are still not giving good enough results, it is likely a sign that you are probably reaching real time rendering limitations, and may actually need to change render engines to something more robust with proper raytracing like Cycles or third party solutions.